The Swans announced the deal on Thursday after the 28-year-old completed a medical before signing a four-year contract at the Liberty Stadium.
After seven years at Las Palmas, the £11m midfielder becomes Swansea's third most expensive signing of all time, only behind Wilfried Bony and £15m Borja Baston.
'One of the main reasons I had to be here was that the manager wanted me', Mesa said upon his arrival, which was announced in a rather unusual way...
An English lesson with a twist... pic.twitter.com/wc7WsGcnHc
— Swansea City AFC (@SwansOfficial) 6 July 2017